ECO 405: HW 2
Sean Fahle
State University of New York at Buffalo
Due date: Friday, February 26, 2015.
You are free to collaborate with your classmates
... [Show More] on this assignment and are encouraged to do so. However, all students must individually write up and submit their own assignment. Assignments are due in class at the beginning of the lecture on the due date. Late assignments may receive no credit.
1. Consider the utility function . Show that (i) x and y are goods, (ii) x and y have diminishing marginal utility, (iii) the indifference curves slope downward, and (iv) the indifference curves are convex. Then, (v) find an equation for and sketch a few representative indifference curves.
2. Mr. Murakami consumes only jazz records and bowls of pasta. If he has more jazzrecords than bowls of pasta, he will always trade 2 jazz records for 1 bowl of pasta. If he has more bowls of pasta than jazz records, he will always give up 1 jazz record for 2 bowls of pasta.
(a) Sketch Mr. Murakami’s indifference curves. Put jazz records on the x axis.
(b) What is his MRS of jazz records for bowls of pasta when he has fewer jazz recordsthan bowls of pasta?
(c) What is his MRS of jazz records for bowls of pasta when he has more jazz recordsthan bowls of pasta?
3. Consider a consumer with preferences given by u(x,y) = xy +6x. Suppose the price of x is px = 2, the price of y is py = 10, and income is I = 600.
(a) Write the Lagrangian for the consumer’s constrained optimization problem.
(b) Write the first order conditions.
(c) Solve for the optimal consumption bundle (x∗,y∗).
√
4. Quasi-linear utility. Consider a consumer with preferences given by u(x,y) = x + y. Suppose the price of x is px = 1 and the price of y is py = 6.
1
(a) Find the optimal consumption bundles at the following three different levels ofincome I = 6, 10, and 20. (That is, solve the problem three times, once for each of these values of income.)
(b) Write an equation for and plot the income expansion path.
5. Kinked Budget Constraint. Mr. Walras has an income equal to 100 which he uses to buy x and y. Suppose px = 2 for the first five units and px = 5 for each additional unit beyond 5. Suppose that py = 5 for each of the first six units and py = 10 for each additional unit beyond 6.
(a) Draw the budget constraint.
(b) If Mr. Walras has the utility function , what is his optimal bundle
(x∗,y∗)? [Show Less]